# Deltoidal hexecontahedron

Deltoidal hexecontahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramm5o3m ()
Conway notationoD
Elements
Faces60 kites
Edges60+60
Vertices12+20+30
Vertex figure20 triangles, 30 squares, 12 pentagons
Measures (edge length 1)
Dihedral angle${\displaystyle \arccos \left(-{\frac {19+8{\sqrt {5}}}{41}}\right)\approx 154.12136^{\circ }}$
Central density1
Number of external pieces60
Level of complexity4
Related polytopes
DualSmall rhombicosidodecahedron
ConjugateGreat deltoidal hexecontahedron
Abstract & topological properties
Flag count480
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexYes
NatureTame

The deltoidal hexecontahedron, also called the strombic hexecontahedron or small lanceal ditriacontahedron, is one of the 13 Catalan solids. It has 60 kites as faces, with 12 order-5, 20 order-3, and 30 order-4 vertices. It is the dual of the uniform small rhombicosidodecahedron.

It can also be obtained as the convex hull of a dodecahedron, an icosahedron, and an icosidodecahedron. If the dodecahedron has unit edge length, the icosahedron's edge length is ${\displaystyle {\frac {7+{\sqrt {5}}}{6}}\approx 1.53934}$ and the icosidodecahedron's edge length is ${\displaystyle {\frac {4-{\sqrt {5}}}{2}}\approx 0.88197}$.

Each face of this polyhedron is a kite with its longer edges ${\displaystyle {\frac {7+{\sqrt {5}}}{6}}\approx 1.53934}$ times the length of its shorter edges. These kites have one angle measuring ${\displaystyle \arccos \left(-{\frac {5+2{\sqrt {5}}}{20}}\right)\approx 118.26868^{\circ }}$, the opposite angle measuring ${\displaystyle \arccos \left({\frac {9{\sqrt {5}}-5}{40}}\right)\approx 67.78301^{\circ }}$, and the other two angles measuring ${\displaystyle \arccos \left({\frac {5-2{\sqrt {5}}}{10}}\right)\approx 86.97416^{\circ }}$.

## Related polytopes

The deltoidal hexecontahedron is topologically equivalent to the rhombic hexecontahedron which has golden rhombi for faces instead of kites.